Double Operator Integrals

This paper is concerned with perturbation formulae of the form∥f(a)−f(b)∥Lp(M,τ)⩽K∥a−b∥ Lp(M,τ) with K>0 being a constant depending on p and f only, where f is a real-valued Lipschitz function and a,b are self-adjoint τ-measurable operators affiliated with a semifinite von Neumann algebra (M,τ), such that the difference a−b belongs to Lp(M,τ), 1<p<∞. In order to treat the situation where the von Neumann algebra M is not necessarily hyperfinite, we first develop an integration theory with respect to finitely additive spectral measures in a Banach space. Applied to product measures this integration theory may be considered as an abstract version of the double operator integrals due to Birman and Solomyak. To describe the class of integrable functions we employ our recent study of multiplier theory in UMD-spaces. Our perturbation formulae extend those of Davies and Birman–Solomyak for the case when M is a hyperfinite I∞-factor (i.e., for the Schatten p-classes). We also discuss analogous perturbation results in the setting of symmetric operator spaces associated with (M,τ).